1,450 research outputs found
Quantum Algorithms and the Fourier Transform
The quantum algorithms of Deutsch, Simon and Shor are described in a way
which highlights their dependence on the Fourier transform. The general
construction of the Fourier transform on an Abelian group is outlined and this
provides a unified way of understanding the efficacy of these algorithms.
Finally we describe an efficient quantum factoring algorithm based on a general
formalism of Kitaev and contrast its structure to the ingredients of Shor's
algorithm.Comment: 18 pages Latex. Submitted to Proceedings of Santa Barbara Conference
on Quantum Coherence and Decoherenc
Counterfactual Computation
Suppose that we are given a quantum computer programmed ready to perform a
computation if it is switched on. Counterfactual computation is a process by
which the result of the computation may be learnt without actually running the
computer. Such processes are possible within quantum physics and to achieve
this effect, a computer embodying the possibility of running the computation
must be available, even though the computation is, in fact, not run. We study
the possibilities and limitations of general protocols for the counterfactual
computation of decision problems (where the result r is either 0 or 1). If p(r)
denotes the probability of learning the result r ``for free'' in a protocol
then one might hope to design a protocol which simultaneously has large p(0)
and p(1). However we prove that p(0)+p(1) never exceeds 1 in any protocol and
we derive further constraints on p(0) and p(1) in terms of N, the number of
times that the computer is not run. In particular we show that any protocol
with p(0)+p(1)=1-epsilon must have N tending to infinity as epsilon tends to 0.
These general results are illustrated with some explicit protocols for
counterfactual computation. We show that "interaction-free" measurements can be
regarded as counterfactual computations, and our results then imply that N must
be large if the probability of interaction is to be close to zero. Finally, we
consider some ways in which our formulation of counterfactual computation can
be generalised.Comment: 19 pages. LaTex, 2 figures. Revised version has some new sections and
expanded explanation
Quantum Coding Theorem for Mixed States
We prove a theorem for coding mixed-state quantum signals. For a class of
coding schemes, the von Neumann entropy of the density operator describing
an ensemble of mixed quantum signal states is shown to be equal to the number
of spin- systems necessary to represent the signal faithfully. This
generalizes previous works on coding pure quantum signal states and is
analogous to the Shannon's noiseless coding theorem of classical information
theory. We also discuss an example of a more general class of coding schemes
which {\em beat} the limit set by our theorem.Comment: Overlap with some unpublished work noted. Limitation clarified. 11
pages, REVTEX, amsfont
Quantum Effects in Algorithms
We discuss some seemingly paradoxical yet valid effects of quantum physics in
information processing. Firstly, we argue that the act of ``doing nothing'' on
part of an entangled quantum system is a highly non-trivial operation and that
it is the essential ingredient underlying the computational speedup in the
known quantum algorithms. Secondly, we show that the watched pot effect of
quantum measurement theory gives the following novel computational possibility:
suppose that we have a quantum computer with an on/off switch, programmed ready
to solve a decision problem. Then (in certain circumstances) the mere fact that
the computer would have given the answer if it were run, is enough for us to
learn the answer, even though the computer is in fact not run.Comment: 10 pages, Latex. For Proceedings of First NASA International
Conference on Quantum Computation and Quantum Communication (Palm Springs,
February 1998
Quantum Information Geometry in the Space of Measurements
We introduce a new approach to evaluating entangled quantum networks using
information geometry. Quantum computing is powerful because of the enhanced
correlations from quantum entanglement. For example, larger entangled networks
can enhance quantum key distribution (QKD). Each network we examine is an
n-photon quantum state with a degree of entanglement. We analyze such a state
within the space of measured data from repeated experiments made by n observers
over a set of identically-prepared quantum states -- a quantum state
interrogation in the space of measurements. Each observer records a 1 if their
detector triggers, otherwise they record a 0. This generates a string of 1's
and 0's at each detector, and each observer can define a binary random variable
from this sequence. We use a well-known information geometry-based measure of
distance that applies to these binary strings of measurement outcomes, and we
introduce a generalization of this length to area, volume and
higher-dimensional volumes. These geometric equations are defined using the
familiar Shannon expression for joint and mutual entropy. We apply our approach
to three distinct tripartite quantum states: the GHZ state, the W state, and a
separable state P. We generalize a well-known information geometry analysis of
a bipartite state to a tripartite state. This approach provides a novel way to
characterize quantum states, and it may have favorable scaling with increased
number of photons.Comment: 21 pages, 7 figure
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